Question

Sejam A(2,1),B(5,-7) e C(-4,-6) os vértices de um triângulo cujo baricentro é G calcule a distância entre A e G

Answer 1

Vamos começar determinando o Baricentro (G):

$\left(x_{_G}~,~y_{_G}\right)~~=~~\left(\dfrac{x_{_A}+x_{_B}+x_{_C}}{3}~,~\\\dfrac{y_{_A}+y_{_B}+y_{_C}}{3}\right)\\\\\\\left(x_{_G}~,~y_{_G}\right)~~=~~\left(\dfrac{2+5+(-4)}{3}~,~\dfrac{1+(-7)+(-6)}{3}\right)\\\\\\\left(x_{_G}~,~y_{_G}\right)~~=~~\left(\dfrac{3}{3}~,~\dfrac{-12}{3}\right)\\\\\\\boxed{\left(x_{_G}~,~y_{_G}\right)~~=~~\left(1~,\,-4\right)}$

Agora podemos determinar a distancia entre A e G:

$Distancia_{\,A,G}~=~\sqrt{\left(x_{_A}-x_{_G}\right)^2~+~\left(y_{_A}-y_{_G}\right)^2}\\\\\\Distancia_{\,A,G}~=~\sqrt{\left(2-1\right)^2~+~\left(1-(-4)\right)^2}\\\\\\Distancia_{\,A,G}~=~\sqrt{\left(1\right)^2~+~\left(5\right)^2}\\\\\\Distancia_{\,A,G}~=~\sqrt{1~+~25}\\\\\\\boxed{Distancia_{\,A,G}~=~\sqrt{26}~~unidades}$